Latex Layout Control

Margin Placing:
http://help-csli.stanford.edu/tex/latex-margins.shtml
Modify the bottom margin distance:
http://www.latex-community.org/forum/viewtopic.php?f=47&t=9531&start=0

Cognitive Radio

Some Basic Concepts from wiki:
http://en.wikipedia.org/wiki/Cognitive_radio

Communication effectively, while avoiding interference with licensed and licensed exempt users. This alternation of parameters is based on the active monitoring of several factors in the external and internal radio environment, such as radio frequency spectrum, user behavior and network state.

Virginia Tech guy defines:"a software defined radio with a cognitive engine brain".

A article link talking about more practical and regulatory considerations.

Some reading to keep up with Cognitive Radio\
June 16, 2011
From Spectral Holes (A blog on Cognitive Radio)
http://spectralholes.blogspot.com/

Latex ! Undefined control sequence

A Guide To Latex (H Kopka, P Daly) 4Ed
P402 Appendix C. Error Messages

! Undefined control sequence.
l.3 The last words appear in \txetbf
{bold face}.
?

! Undefined control sequence, meaning that an unknown command name (control sequence) was the cause of the error. Next comes a pair of text lines, the first of which is prefixed with 1.3, meaning that the error occurred in 'line 3' of the input text. The error itself was encountered at the last symbol printed in the upper line. The lower line shows the continuation of the input line being processed when the error was found, here the words {bold face}. Before continuing, TEX waits for a reaction from the user, as indicated by the question mark in the last line of the message.

For my case:

! Undefined control sequence.

l.9 A random vector \underscore

                               {x}:

should be changed to \underline{x}

fixed

Latex Symbols and other resource

Unicode Standard Documents:
http://unicode.org/

Delta Equal
http://unicode.org/charts/PDF/U2200.pdf

Short Math Guide for Latex:
ftp://ftp.ams.org/ams/doc/amsmath/short-math-guide.pdf

\triangleq #delta equal to

Consulting The Comprehensive Latex Symbol List (Pakin):

http://www.ctan.org/tex-archive/info/symbols/comprehensive/

How to displaying a formula:
(Useful link for Wikipedia):
http://en.wikipedia.org/wiki/Help:Displaying_a_formula

Kronecker Product
\otimes

FAQ for Latex
http://www.tug.org/pracjourn/2006-1/schmidt/schmidt.pdf

Beamer by Example:
http://tug.org/pracjourn/2005-4/mertz/mertz.pdf

several examples for report:
https://engineering.purdue.edu/ECN/Support/KB/Docs/LaTeXSampleTemplateF
also contains CV example
http://www.tedpavlic.com/post_homework_tex_example.php
http://links.tedpavlic.com/tex/homework.tex
http://maths.dur.ac.uk/Ug/projects/resources/latex/report/

Install Latex in Linux

http://linuxandfriends.com/2009/10/06/install-latex-in-ubuntu-linux/

Setting up Latex
http://www-lmmb.ncifcrf.gov/~toms/latexforbeginners.html

ldd
print shared library dependencies.
Normally, using under /usr/bin /usr/shared /usr/lib path

LaTex Beamer Matrix
http://deic.uab.es/~iblanes/beamer_gallery/index.html
http://www.hartwork.org/beamer-theme-matrix/

Tex Showcase
http://www.tug.org/texshowcase/

LaTex Command Glossary:
http://en.wikibooks.org/wiki/LaTeX/Command_Glossary

Wiki Books:
http://en.wikibooks.org

Other useful resources:
A Guide To Latex and Electronic Publishing

Perturbation Theory

First-order non-singular perturbation theory

This section develops, in simplified terms, the general theory for the perturbative solution to a differential equation to the first order. To keep the exposition simple, a crucial assumption is made: that the solutions to the unperturbed system are not degenerate, so that the perturbation series can be inverted. There are ways of dealing with the degenerate (or singular) case; these require extra care.

Suppose one wants to solve a differential equation of the form

Dg(x) = λg(x)

where D is some specific differential operator, and λ is an eigenvalue. Many problems involving ordinary or partial differential equations can be cast in this form. It is presumed that the differential operator can be written in the form

where  is presumed to be small, and that furthermore, the complete set of solutions for D⁽⁰⁾ are known. That is,one has a set of solutions , labelled by some arbitrary index n, such that

.

Fredholm Integral Equation of the First Kind

Equation of the first kind

Integral equations, most generally, are common and take many specific forms (Fourier, Laplace, Hankel, etc.). They each differ in their kernels (defined below). What is distinctive about Fredholm integral equations is that they are integral equations in which the integration limits are constants (they do not include the variable). This is contrast to Volterra integral equations.

A homogeneous Fredholm equation of the first kind is written as:

General theory

The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel K is a compact operator, known as the Fredholm operator. Compactness may be shown by invoking equicontinuity. As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.

[edit]Applications

Fredholm equations arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian. They also commonly arise in linear forward modeling and inverse problems.

Perturbative inversion methods

Perturbative inversion methods for obtaining bottom geoacoustic parameters

Attention is focused primarily on the modal eigenvalue inverse problem for which the theory for determining the compressional  wave speed, compressional wave attenuation, and density is developed in detail. Properties of this technique are studied using synthetic data and include investigations of the dependence of the results on acoustics frequency, number of modes excited, and partial a priori knowledge of the bottom.

Inversion Problem:
The required input data are trapped mode eigenvalues for one or more frequencies, the group velocity dispersion curves for one or more modes, or the cw pressure field versus range (complex field or magnitude only) .

A great explanation of inverse problem:
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. For example, if we have measurements of the Earth's gravity field, then we might ask the question: "given the data that we have available, what can we say about the density distribution of the Earth in that area?" The solution to this problem (i.e. the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics. Inverse problems arise in many branches of science and mathematics, including: computer vision, machine learning, statistics, statistical inference, geophysics, medical imaging (such as computed axial tomographyand EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.


(Linear Inverse Theory)

The objective of an inverse problem is to find the best model, m, such that (at least approximately)

where G is an operator describing the explicit relationship between the observed data, d, and the model parameters. In various contexts, the operator G is called forward operator,observation operator, or observation function. In the most general context, G represents the governing equations that relate the model parameters to the observed data (i.e. the governing physics).

Mathematical

One central example of a linear inverse problem is provided by a Fredholm first kind integral equation.

Check Fredholm first kind integral equation in next several posts
For sufficiently smooth g the operator defined above is compact on reasonable Banach spaces such as L^p spaces. Even if the mapping is injective its inverse will not be continuous. (However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (i.e. continuous).) Thus small errors in the data d are greatly amplified in the solution m. In this sense the inverse problem of inferring m from measured d is ill-posed.


Numerical Scheme (quadrature scheme):
Simpson's Rule
http://en.wikipedia.org/wiki/Simpson's_rule
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:





2 Regularization Method

Regularization:
http://en.wikipedia.org/wiki/Regularization_(mathematics)
Regularization of Inverse Problem
http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-0-7923-4157-4

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that

1. A solution exists
2. The solution is unique
3. The solution depends continuously on the data, in some reasonable topology.

http://en.wikipedia.org/wiki/Ill-posed_problem

In mathematics and statistics, particularly in the fields of machine learning and inverse problems, regularization involves introducing additional information in order to solve an ill-posed problem or to prevent overfitting. This information is usually of the form of a penalty for complexity, such as restrictions for smoothness or bounds on the vector space norm.

Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed.

Such continuum problems must often be discretized in order to obtain a numerical solution. While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data. Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. An ill-conditioned problem is indicated by a large condition number.

If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization and Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.

Other Keywords:
Normed Vector Space/Vector Space
minimum norm solution
A MatLab Program: Least squires with solving minimum norm solution
http://www.mathworks.com/matlabcentral/fileexchange/17474


Smooth Function:
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.